Question: Let $a,$ $b,$ $c,$ $z$ be complex numbers such that $|a| = |b| = |c| > 0$ and
\[az^2 + bz + c = 0.\]Find the largest possible value of $|z|.$
Answer: Let $r = |a| = |b| = |c|.$  We can re-arrange $az^2 + bz + c = 0$ as
\[az^2 = -bz - c.\]By the Triangle Inequality,
\[|az^2| = |-bz - c| \le |bz| + |c|,\]so $|a||z|^2 \le |b||z| + |c|,$ or $r|z|^2 \le r|z| + r.$  Then
\[|z|^2 \le |z| + 1,\]so $|z|^2 - |z| - 1 \le 0.$  This factors as
\[\left( |z| - \frac{1 - \sqrt{5}}{2} \right) \left( |z| - \frac{1 + \sqrt{5}}{2} \right) \le 0,\]so $|z| \le \frac{1 + \sqrt{5}}{2}.$

The numbers $a = 1,$ $b = -1,$ $c = -1,$ and $z = \frac{1 + \sqrt{5}}{2}$ satisfy the given conditions, so the largest possible value of $|z|$ is $\boxed{\frac{1 + \sqrt{5}}{2}}.$